 Reference identity in algebraic uncertainty cancellation

Reference identity in algebraic uncertainty cancellation asks whether two appearances of a reference are the same measured quantity or merely readings given the same descriptive name. If a transfer is $D=B-A_1$ but a correction subtracts $A_2$, thenThe last difference vanishes identically only when the same reference value is reused. Separate acquisitions may be correlated, independent, or affected by drift; sharing a label does not decide their covariance.

For independent $T$ and $B$, also independent of the reference pair, propagation givesWith equal reference standard uncertainties $s$ and correlation coefficient $\rho$, the residual reference contribution is $2s^2(1-\rho )$. It is zero for identical reused values, $2s^2$ for independent readings, and is not generally zero just because the displayed numbers agree. The GUM requires significant correlations from common inputs to be included.^[ref]

An original acquisition model separates what cancels:Here $C$ is a common uncertain calibration contribution, while ${ϵ}_1,{ϵ}_2$ are independent repeat errors. Their difference removes both $A_0$ and $C$, but leaves ${ϵ}_1-{ϵ}_2$. If $u(C)=3$ mg and each repeat error has standard deviation 2 mg, each reading has variance $13\mathrm{m}{\mathrm{g}}^2$ and their covariance is $9\mathrm{m}{\mathrm{g}}^2$. The difference therefore has variance $13+13-18=8\mathrm{m}{\mathrm{g}}^2$, not zero.

Adding independent $u(T)=1$ mg and $u(B)=3$ mg gives $u^2(Y)=18\mathrm{m}{\mathrm{g}}^2$. Reusing exactly one recorded reference value in both places instead gives $u^2(Y)=10\mathrm{m}{\mathrm{g}}^2$. These constructed cases have different data-reduction histories despite similar-looking equations.

Reference lineage in covariance models records acquisition identifiers, shared calibration inputs and coefficients in each derived quantity. A calibration change or time-dependent drift must appear in that model if significant. Algebraic simplification is valuable only when it preserves the identity of the underlying inputs; renaming two acquisitions with the same symbol does not make their errors cancel.